ScalingStacks

The map of spaces \[\begin{aligned} \mathrm{Im}\left(S_n \rightarrow\vphantom{\mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]_{/\mathcal D}}}\right. & \left. \mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]_{/\mathcal D}} \left( \mathcal C^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\right)\right) \\ &\longrightarrow \mathrm{Im}\left(S_n \rightarrow\mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]_{/\mathcal D}}\left(\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathcal B)^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C) \right) \right) \end{aligned}\] is an equivalence.